Thanks to overwhelming consumer demand, Europe’s pre-eminent cultural attractions are about to get bigger, unrestrained by small, outmoded buildings – or even international borders.

The British Museum in London, which opened in 1759 in Montagu House, a 17th-century mansion on the site of the current museum, welcomed 5,000 visitors in its first year. In 2013, it received 6.5 million people. To accommodate these growing crowds, the adjoining World Conservation and Exhibitions Centre opens this month. The £135 million extension, designed by Rogers Stirk Harbour+Partners, will provide nine floors of office and exhibition space and should ensure a comfortable visitor experience for decades, if not centuries, to come.

But the British Museum isn’t the only London attraction expanding to accommodate the capital’s voracious consumers of culture. Tate Modern, which was expected to receive 2.5 million visitors in its first year but actually attracted five million, is to open an extension in 2016, designed by Herzog & de Meuron. A dramatic, swerving brick tower, it will reach the same height as the former power station’s chimney.

Meanwhile, in South Kensington, the Victoria & Albert Museum is developing several new spaces. The most notable addition, designed by Amanda Levete Architects, is the Exhibition Road Building Project, an underground gallery with a courtyard. It will have its own entrance on Exhibition Road, and opens in 2017.

Housed in a converted banana warehouse, the Design Museum in Shad Thames, near Tower Bridge, lacks the space to expand, so it is taking more drastic measures. Later this year it will relocate to the former Commonwealth Institute on Kensington High Street; John Pawson, most recently recognised for his work on Living Architecture's Life House in Wales and Augsburg’s Moritzkirche, is remodelling the building’s interiors.

But some cultural institutions have gone even further in their expansion plans. The V&A is also opening a site in Dundee in 2015; designed by Kengo Kuma, the attraction will anchor the regeneration of the city's waterfront and is set to be an epicentre for the celebration of design in Scotland. The Louvre in Paris opened a branch in the French city of Lens in 2012. More controversially, Louvre Abu Dhabi is set to open in 2015. Designed by Jean Nouvel and developed with guidance from Agence France-Muséums, the vast building will be a focal point for the cultural district being developed on Saadiyat Island. It will display a collection “enriched by loans” not only from the Louvre, but also from the Musée d’Orsay and Centre Pompidou in Paris. Guggenheim Abu Dhabi, designed by Frank Gehry, will open in 2017.

Whether fans of French art will forgo Paris for museums on the Persian Gulf remains to be seen, but Saadiyat Island will be a significant destination when works are completed in 2020. The development will include the striking Zayed National Mulseum, designed by Norman Foster, and a sinuous performing arts centre conceived by Zaha Hadid.

In this age of big-budget, big-brand cultural centres, a new generation of attractions is about to push the boundaries even further.